X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fcsuba%2Ffwd.ma;h=9635a44a0c2eab1d6296ab957bf0082393a6dbf2;hb=076f639446efce8d8cf83dcf7ca40b4376fc8c36;hp=6a34ae7e78cffd4f1b3685fe7d295ff55679032e;hpb=d0982534aee06a30f91a06d2f3e82834b132a3d3;p=helm.git diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd.ma index 6a34ae7e7..9635a44a0 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/csuba/fwd.ma @@ -14,7 +14,7 @@ (* This file was automatically generated: do not edit *********************) -include "csuba/defs.ma". +include "LambdaDelta-1/csuba/defs.ma". theorem csuba_gen_abbr: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g @@ -22,54 +22,54 @@ theorem csuba_gen_abbr: (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) \def \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g (CHead d1 (Bind Abbr) u) c)).(let H0 \def (match H in csuba return -(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 -(CHead d1 (Bind Abbr) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C -c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with -[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind -Abbr) u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort -n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abbr) -u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c -(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) | -(csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) -(CHead d1 (Bind Abbr) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k -u0) (CHead d1 (Bind Abbr) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) -in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) -(CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in (eq_ind C d1 (\lambda (c0: -C).((eq K k (Bind Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to -((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind -Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead -c2 k0 u0) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead -d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq -T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Abbr) t) c) \to -((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) -u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2 -(Bind Abbr) u) c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c0: -C).((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind -Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1 -c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 -(Bind Abbr) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Abbr) -H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a -H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 -(Bind Abbr) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5 -\def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H3) in (False_ind ((eq C -(CHead c2 (Bind Abbr) u0) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g -a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 -(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))]) -in (H0 (refl_equal C (CHead d1 (Bind Abbr) u)) (refl_equal C c))))))). +(csuba g (CHead d1 (Bind Abbr) u) c)).(insert_eq C (CHead d1 (Bind Abbr) u) +(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq +C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda +(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda +(c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C +c1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda +(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(let H2 +\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow +False])) I (CHead d1 (Bind Abbr) u) H1) in (False_ind (ex2 C (\lambda (d2: +C).(eq C (CSort n) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0) +(CHead d1 (Bind Abbr) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) in ((let H5 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k +u0) (CHead d1 (Bind Abbr) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | +(CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) +in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r +T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2 +(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Abbr) +(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2 +(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C +c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 +c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) +u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 +(refl_equal C (CHead c2 (Bind Abbr) u)) H10))) k H7) u0 H6)))) H5)) +H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 +c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 +d2)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g +a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C +(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C +(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B +return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow +True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 +(Bind Abbr) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 +(Bind Abbr) u0) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) +H6)))))))))))) y c H0))) H))))). theorem csuba_gen_void: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g @@ -77,54 +77,54 @@ theorem csuba_gen_void: (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) \def \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g (CHead d1 (Bind Void) u) c)).(let H0 \def (match H in csuba return -(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 -(CHead d1 (Bind Void) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C -c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with -[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind -Void) u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort -n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void) -u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c -(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) | -(csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) -(CHead d1 (Bind Void) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k -u0) (CHead d1 (Bind Void) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H1) -in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) -(CHead c1 k u0) (CHead d1 (Bind Void) u) H1) in (eq_ind C d1 (\lambda (c0: -C).((eq K k (Bind Void)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to -((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) -u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind -Void))).(eq_ind K (Bind Void) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead -c2 k0 u0) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead -d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq -T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Void) t) c) \to -((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) -u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2 -(Bind Void) u) c)).(eq_ind C (CHead c2 (Bind Void) u) (\lambda (c0: -C).((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind -Void) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1 -c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Void) u) (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 -(Bind Void) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Void) -H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a -H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 -(Bind Void) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5 -\def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead d1 (Bind Void) u) H3) in (False_ind ((eq C -(CHead c2 (Bind Abbr) u0) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g -a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))]) -in (H0 (refl_equal C (CHead d1 (Bind Void) u)) (refl_equal C c))))))). +(csuba g (CHead d1 (Bind Void) u) c)).(insert_eq C (CHead d1 (Bind Void) u) +(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq +C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda +(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda +(c1: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C +c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda +(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2 +\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow +False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2: +C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 +d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 +d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0) +(CHead d1 (Bind Void) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H3) in ((let H5 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k +u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | +(CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H3) +in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r +T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2 +(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Void) +(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2 +(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C +c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 +d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 +c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Void) +u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 +(refl_equal C (CHead c2 (Bind Void) u)) H10))) k H7) u0 H6)))) H5)) +H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 +c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda +(d2: C).(eq C c2 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 +d2)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g +a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C +(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C +(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B +return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow +True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 +(Bind Void) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2 +(Bind Abbr) u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))) +H6)))))))))))) y c H0))) H))))). theorem csuba_gen_abst: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g @@ -137,122 +137,117 @@ a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))) \def \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(let H0 \def (match H in csuba -return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C -c0 (CHead d1 (Bind Abst) u1)) \to ((eq C c1 c) \to (or (ex2 C (\lambda (d2: +(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(insert_eq C (CHead d1 (Bind Abst) +u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a))))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: -(eq C (CSort n) (CHead d1 (Bind Abst) u1))).(\lambda (H1: (eq C (CSort n) -c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) -\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H0) in (False_ind ((eq C -(CSort n) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) +A).(arity g d2 u2 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g y +c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind +Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) +C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) -(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) -H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead -c1 k u) (CHead d1 (Bind Abst) u1))).(\lambda (H2: (eq C (CHead c2 k u) -c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) -(CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) in ((let H4 \def (f_equal C K -(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) -\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead d1 -(Bind Abst) u1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in -C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) in (eq_ind C -d1 (\lambda (c0: C).((eq K k (Bind Abst)) \to ((eq T u u1) \to ((eq C (CHead -c2 k u) c) \to ((csuba g c0 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c -(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind -Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 -d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc -g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))))) (\lambda (H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) -(\lambda (k0: K).((eq T u u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1 -c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) +(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) +u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H1) in (False_ind (or (ex2 C +(\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u1))) (\lambda (d2: +C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(eq C (CSort n) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))) (\lambda -(H7: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 (Bind Abst) -t) c) \to ((csuba g d1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 -(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abst) u1) c)).(eq_ind C (CHead -c2 (Bind Abst) u1) (\lambda (c0: C).((csuba g d1 c2) \to (or (ex2 C (\lambda -(d2: C).(eq C c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 +(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H2)))) +(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda +(H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: +C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) +(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: -A).(arity g d2 u2 a)))))))) (\lambda (H9: (csuba g d1 c2)).(or_introl (ex2 C -(\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abbr) -u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g -a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 -a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead -d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C -(CHead c2 (Bind Abst) u1)) H9))) c H8)) u (sym_eq T u u1 H7))) k (sym_eq K k -(Bind Abst) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst -c1 c2 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) -t) (CHead d1 (Bind Abst) u1))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u) -c)).((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) -(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) H3) in ((let H6 \def -(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind -Abst) t) (CHead d1 (Bind Abst) u1) H3) in (eq_ind C d1 (\lambda (c0: C).((eq -T t u1) \to ((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g c0 c2) \to -((arity g c0 t (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda -(d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity -g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: -A).(arity g d2 u2 a0)))))))))))) (\lambda (H7: (eq T t u1)).(eq_ind T u1 -(\lambda (t0: T).((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g d1 c2) \to -((arity g d1 t0 (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda -(d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity -g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: -A).(arity g d2 u2 a0))))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abbr) u) -c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c0: C).((csuba g d1 c2) \to -((arity g d1 u1 (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda -(d2: C).(eq C c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 -d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 -(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity -g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: -A).(arity g d2 u2 a0)))))))))) (\lambda (H9: (csuba g d1 c2)).(\lambda (H10: -(arity g d1 u1 (asucc g a))).(\lambda (H11: (arity g c2 u a)).(or_intror (ex2 -C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abst) u1))) -(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda +A).(arity g d2 u2 a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: +(eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(let H4 \def (f_equal C C +(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1 +(Bind Abst) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in +C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) +\Rightarrow k0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H3) in ((let H6 +\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) +with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) +(CHead d1 (Bind Abst) u1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda +(H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2: +C).(eq C (CHead c2 k t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g +d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C +(CHead c2 k t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: +T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: +T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda +(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (eq_ind_r K (Bind Abst) +(\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u1) (CHead d2 +(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 k0 u1) (CHead d2 +(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g +d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 +u2 a))))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 +(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind +Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) H2 +d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 +H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) +u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: +A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda +(a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 +(Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) +c2 (refl_equal C (CHead c2 (Bind Abst) u1)) H10)))) k H7) u H6)))) H5)) +H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C +(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba +g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq +C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: +A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda +(a: A).(arity g d2 u2 a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda +(H3: (arity g c1 t (asucc g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u +a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) +u1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda +(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) +(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) H5) in ((let H7 \def +(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind +Abst) t) (CHead d1 (Bind Abst) u1) H5) in (\lambda (H8: (eq C c1 d1)).(let H9 +\def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 u1 H7) in +(let H10 \def (eq_ind C c1 (\lambda (c0: C).(arity g c0 u1 (asucc g a))) H9 +d1 H8) in (let H11 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1 +(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind +Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: +C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) +(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))) +H2 d1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 +d1 H8) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) +(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A +(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) +u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda +(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: +A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda +(a0: A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 -a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: -A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: -C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: -C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))) c2 u a -(refl_equal C (CHead c2 (Bind Abbr) u)) H9 H10 H11))))) c H8)) t (sym_eq T t -u1 H7))) c1 (sym_eq C c1 d1 H6))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C -(CHead d1 (Bind Abst) u1)) (refl_equal C c))))))). +a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H12 H10 H4)))))))) +H6)))))))))))) y c H0))) H))))). theorem csuba_gen_flat: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall @@ -261,57 +256,57 @@ C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) \def \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(let H0 \def (match H -in csuba return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 -c1)).((eq C c0 (CHead d1 (Flat f) u1)) \to ((eq C c1 c) \to (ex2_2 C T +(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(insert_eq C (CHead +d1 (Flat f) u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) with [(csuba_sort n) -\Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Flat f) u1))).(\lambda -(H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: -C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow -True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Flat f) u1) H0) in -(False_ind ((eq C (CSort n) c) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d1 d2))))) H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: -(eq C (CHead c1 k u) (CHead d1 (Flat f) u1))).(\lambda (H2: (eq C (CHead c2 k -u) c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e in C return -(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow -t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in ((let H4 \def (f_equal C K -(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) -\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead d1 -(Flat f) u1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in C -return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) -\Rightarrow c0])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in (eq_ind C d1 -(\lambda (c0: C).((eq K k (Flat f)) \to ((eq T u u1) \to ((eq C (CHead c2 k -u) c) \to ((csuba g c0 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d1 d2))))))))) (\lambda (H6: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda -(k0: K).((eq T u u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1 c2) \to -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))) (\lambda (H7: -(eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 (Flat f) t) c) \to -((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c -(CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))))))) (\lambda (H8: (eq C (CHead c2 (Flat f) u1) c)).(eq_ind C (CHead c2 -(Flat f) u1) (\lambda (c0: C).((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d1 d2)))))) (\lambda (H9: (csuba g d1 -c2)).(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat -f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 -d2))) c2 u1 (refl_equal C (CHead c2 (Flat f) u1)) H9)) c H8)) u (sym_eq T u -u1 H7))) k (sym_eq K k (Flat f) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 -H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C -(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(\lambda (H4: (eq C (CHead -c2 (Bind Abbr) u) c)).((let H5 \def (eq_ind C (CHead c1 (Bind Abst) t) -(\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow -False])])) I (CHead d1 (Flat f) u1) H3) in (False_ind ((eq C (CHead c2 (Bind -Abbr) u) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity -g c2 u a) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))) H5)) -H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Flat f) u1)) (refl_equal C -c)))))))). +(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (\lambda (y: C).(\lambda (H0: +(csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 +(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) +u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) H2)))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1 +(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) +(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in ((let H5 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k +u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in +(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1 +(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 +k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (let H9 \def (eq_ind C c1 +(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda +(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d1 d2)))))) H2 d1 H8) in (let H10 \def (eq_ind C +c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat f) u1) (CHead d2 (Flat f) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C +(CHead c2 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1 +(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 +d2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g +a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C +(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C +(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat +_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C +T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 +(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) +H6)))))))))))) y c H0))) H)))))). theorem csuba_gen_bind: \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall @@ -320,92 +315,80 @@ B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) \def \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda -(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(let H0 \def -(match H in csuba return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csuba -? c c0)).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 c2) \to (ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind -b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 -e2)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) -(CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def -(eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) c2) \to -(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 +(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(insert_eq C +(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c c2)) (\lambda (_: +C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e1 e2)))))) H2)) H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow -(\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2: -(eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e -in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) -\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H4 \def -(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with -[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) -(CHead e1 (Bind b1) v1) H1) in ((let H5 \def (f_equal C C (\lambda (e: -C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | -(CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in -(eq_ind C e1 (\lambda (c: C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((eq C -(CHead c0 k u) c2) \to ((csuba g c c0) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) -(\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind b1) (\lambda (k0: K).((eq T -u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csuba g e1 c0) \to (ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind -b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 -e2))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C -(CHead c0 (Bind b1) t) c2) \to ((csuba g e1 c0) \to (ex2_3 B C T (\lambda -(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) -v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 -e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1) c2)).(eq_ind C (CHead -c0 (Bind b1) v1) (\lambda (c: C).((csuba g e1 c0) \to (ex2_3 B C T (\lambda -(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) -(\lambda (H9: (csuba g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c: -C).(csuba g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b1) v1) H8) in -(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C -(CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda -(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c0 v1 (refl_equal C (CHead c0 -(Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind b1) -H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c0 H0 t a -H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead e1 -(Bind b1) v1))).(\lambda (H4: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H5 -\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 -(Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def (f_equal C B -(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) -\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda -(_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) -(CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H7 \def +T).(csuba g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csuba g y +c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind +b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq +C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) +v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda +(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda +(e2: C).(\lambda (_: T).(csuba g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda +(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 +(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda +(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (k: K).(\lambda (u: +T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with -[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind -Abst) t) (CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c: C).((eq B -Abst b1) \to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba -g c c0) \to ((arity g c t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind -b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 -e2)))))))))))) (\lambda (H8: (eq B Abst b1)).(eq_ind B Abst (\lambda (_: -B).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0) -\to ((arity g e1 t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind +[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) +(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: +C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | +(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3) +in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) +(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind +b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t) +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1) +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c +(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H8) in (let +H10 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H8) in +(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C +(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda +(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 +(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: +C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 +(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (t: +T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u: +T).(\lambda (_: (arity g c3 u a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) +t) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match +e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ +_) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in +((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: +C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k +in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) +\Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in +((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead +c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abst +b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda (t0: +T).(arity g c1 t0 (asucc g a))) H3 v1 H8) in (let H12 \def (eq_ind C c1 +(\lambda (c: C).(arity g c v1 (asucc g a))) H11 e1 H10) in (let H13 \def +(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C +T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 -e2))))))))))) (\lambda (H9: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C -(CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0) \to ((arity g e1 t0 (asucc -g a)) \to ((arity g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: -B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) (\lambda (H10: -(eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) -(\lambda (c: C).((csuba g e1 c0) \to ((arity g e1 v1 (asucc g a)) \to ((arity -g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: -T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e1 e2))))))))) (\lambda (H11: (csuba g e1 -c0)).(\lambda (_: (arity g e1 v1 (asucc g a))).(\lambda (_: (arity g c0 u -a)).(let H14 \def (eq_ind_r C c2 (\lambda (c: C).(csuba g (CHead e1 (Bind b1) -v1) c)) H (CHead c0 (Bind Abbr) u) H10) in (let H15 \def (eq_ind_r B b1 -(\lambda (b: B).(csuba g (CHead e1 (Bind b) v1) (CHead c0 (Bind Abbr) u))) -H14 Abst H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda -(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) Abbr c0 u -(refl_equal C (CHead c0 (Bind Abbr) u)) H11)))))) c2 H10)) t (sym_eq T t v1 -H9))) b1 H8)) c1 (sym_eq C c1 e1 H7))) H6)) H5)) H4 H0 H1 H2)))]) in (H0 -(refl_equal C (CHead e1 (Bind b1) v1)) (refl_equal C c2)))))))). +e2))))))) H2 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c: C).(csuba g c +c3)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 +(CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H13 Abst H9) in +(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C +(CHead c3 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda +(e2: C).(\lambda (_: T).(csuba g e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 +(Bind Abbr) u)) H14))))))))) H7)) H6)))))))))))) y c2 H0))) H)))))). theorem csuba_gen_abst_rev: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c @@ -413,62 +396,54 @@ theorem csuba_gen_abst_rev: Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))) \def \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g c (CHead d1 (Bind Abst) u))).(let H0 \def (match H in csuba return -(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c) -\to ((eq C c1 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C c -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with -[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1: -(eq C (CSort n) (CHead d1 (Bind Abst) u))).(eq_ind C (CSort n) (\lambda (c0: -C).((eq C (CSort n) (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq -C c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda -(H2: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H3 \def (eq_ind C (CSort -n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abst) -u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind -Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head -c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda -(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u))).(eq_ind C (CHead c1 k -u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u)) \to -((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k -u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C T (\lambda (e: +(csuba g c (CHead d1 (Bind Abst) u))).(insert_eq C (CHead d1 (Bind Abst) u) +(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq +C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda +(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda +(c1: C).((eq C c1 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C +c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda +(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2 +\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow +False])) I (CHead d1 (Bind Abst) u) H1) in (False_ind (ex2 C (\lambda (d2: +C).(eq C (CSort n) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda +(d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0) +(CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H5 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k +u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) -in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) -(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 -(Bind Abst) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Abst)) \to -((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1 -k u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))) -(\lambda (H7: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: -K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead -c1 k0 u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))) -(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to -(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) -u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1 -d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2 -(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 -(Bind Abst) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Abst) H7))) -c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a -H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) -c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) -u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2 -(Bind Abbr) u0) (CHead d1 (Bind Abst) u)) \to ((csuba g c1 c2) \to ((arity g -c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c0 -(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda -(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def -(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead d1 (Bind Abst) u) H5) in (False_ind ((csuba g -c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C -(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) u))) -(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0 -(refl_equal C c) (refl_equal C (CHead d1 (Bind Abst) u)))))))). +in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r +T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Abst) +(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2 +(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C +c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda +(d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 +c0)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) +u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 +(refl_equal C (CHead c1 (Bind Abst) u)) H10))) k H7) u0 H6)))) H5)) +H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 +c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda +(d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 +d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g +a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C +(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def (eq_ind C +(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B +return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow +False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead +d1 (Bind Abst) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c1 +(Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) +H6)))))))))))) c y H0))) H))))). theorem csuba_gen_void_rev: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c @@ -476,62 +451,54 @@ theorem csuba_gen_void_rev: Void) u))) (\lambda (d2: C).(csuba g d2 d1))))))) \def \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: -(csuba g c (CHead d1 (Bind Void) u))).(let H0 \def (match H in csuba return -(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c) -\to ((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c -(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with -[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1: -(eq C (CSort n) (CHead d1 (Bind Void) u))).(eq_ind C (CSort n) (\lambda (c0: -C).((eq C (CSort n) (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq -C c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda -(H2: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H3 \def (eq_ind C (CSort -n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) -\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void) -u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind -Void) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head -c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda -(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Void) u))).(eq_ind C (CHead c1 k -u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Void) u)) \to -((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Void) -u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k -u0) (CHead d1 (Bind Void) u))).(let H4 \def (f_equal C T (\lambda (e: +(csuba g c (CHead d1 (Bind Void) u))).(insert_eq C (CHead d1 (Bind Void) u) +(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq +C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda +(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda +(c1: C).((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C +c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda +(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2 +\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow +False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2: +C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 +d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 +c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda +(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 +d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0) +(CHead d1 (Bind Void) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H5 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k +u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) -in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda -(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) -(CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 -(Bind Void) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Void)) \to -((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1 -k u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))))) -(\lambda (H7: (eq K k (Bind Void))).(eq_ind K (Bind Void) (\lambda (k0: -K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead -c1 k0 u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) -(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to -(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) t) (CHead d2 (Bind Void) -u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1 -d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u) (CHead d2 -(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 -(Bind Void) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Void) H7))) -c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a -H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) -c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) -u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2 -(Bind Abbr) u0) (CHead d1 (Bind Void) u)) \to ((csuba g c1 c2) \to ((arity g -c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c0 -(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda -(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def -(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e in C return -(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) -\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) -\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow -True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) -\Rightarrow False])])) I (CHead d1 (Bind Void) u) H5) in (False_ind ((csuba g -c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C -(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u))) -(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0 -(refl_equal C c) (refl_equal C (CHead d1 (Bind Void) u)))))))). +in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r +T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2 +(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Void) +(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2 +(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C +c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda +(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 +d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1 +c0)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) +u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 +(refl_equal C (CHead c1 (Bind Void) u)) H10))) k H7) u0 H6)))) H5)) +H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1 +c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda +(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 +d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g +a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C +(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C +(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_: +C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match +k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B +return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow +False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead +d1 (Bind Void) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c1 +(Bind Abst) t) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))) +H6)))))))))))) c y H0))) H))))). theorem csuba_gen_abbr_rev: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c @@ -544,132 +511,117 @@ a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))) \def \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(let H0 \def (match H in csuba -return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C -c0 c) \to ((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: +(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(insert_eq C (CHead d1 (Bind Abbr) +u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a: A).(arity g d1 u1 a))))))))))) with [(csuba_sort n) \Rightarrow (\lambda -(H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) -u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort n) (CHead d1 (Bind +(a: A).(arity g d1 u1 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g c +y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))) -(\lambda (H2: (eq C (CSort n) (CHead d1 (Bind Abbr) u1))).(let H3 \def -(eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead d1 (Bind Abbr) u1) H2) in (False_ind (or (ex2 C (\lambda -(d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g -d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C -(CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: -T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H3))) c H0 H1))) | (csuba_head -c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda -(H2: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead c1 k -u) (\lambda (c0: C).((eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1)) \to -((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) -(\lambda (H3: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(let H4 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u) -(CHead d1 (Bind Abbr) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: -C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) -in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) -(CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in (eq_ind C d1 (\lambda (c0: -C).((eq K k (Bind Abbr)) \to ((eq T u u1) \to ((csuba g c1 c0) \to (or (ex2 C -(\lambda (d2: C).(eq C (CHead c1 k u) (CHead d2 (Bind Abbr) u1))) (\lambda -(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C (CHead c1 k u) (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))) -(\lambda (H7: (eq K k (Bind Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: -K).((eq T u u1) \to ((csuba g c1 d1) \to (or (ex2 C (\lambda (d2: C).(eq C -(CHead c1 k0 u) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 -d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C -(CHead c1 k0 u) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: -T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: +(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) +u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead d1 (Bind Abbr) u1) H1) in (False_ind (or (ex2 C +(\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: +C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda +(_: A).(eq C (CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H2)))) (\lambda +(c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C +c2 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 +a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) +(CHead d1 (Bind Abbr) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match +e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H5 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k +u) (CHead d1 (Bind Abbr) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) +in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r +T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead +d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 k t) (CHead d2 (Bind +Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 +d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 +(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 +u1 a))))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex2 C (\lambda (d2: +C).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba +g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq +C (CHead c1 k0 u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda +(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda -(_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) (\lambda (H8: (eq T u -u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (or (ex2 C (\lambda -(d2: C).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda +(_: T).(\lambda (a: A).(arity g d1 u1 a))))))) (let H9 \def (eq_ind C c2 +(\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda +(d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 +(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: +A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: +A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda +(a: A).(arity g d1 u1 a)))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 +(\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (or_introl (ex2 C (\lambda +(d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: -T).(\lambda (_: A).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abst) +T).(\lambda (_: A).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) +u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) +(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g +a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) +(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind +Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 +(Bind Abbr) u1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 +(CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 +(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda +(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 -a)))))))) (\lambda (H9: (csuba g c1 d1)).(or_introl (ex2 C (\lambda (d2: -C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: -C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda -(_: A).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda -(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: -C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda -(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C -(\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) -(\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind Abbr) u1)) -H9))) u (sym_eq T u u1 H8))) k (sym_eq K k (Bind Abbr) H7))) c2 (sym_eq C c2 -d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) -\Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4: -(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead -c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1 -(Bind Abbr) u1)) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to -((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind -Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: -C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2))))) -(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda -(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g a0))))) -(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 a0))))))))))) -(\lambda (H5: (eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(let -H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) -with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2 -(Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in ((let H7 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) +a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc +g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C +(CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(let H6 \def (f_equal C +C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u) -(CHead d1 (Bind Abbr) u1) H5) in (eq_ind C d1 (\lambda (c0: C).((eq T u u1) -\to ((csuba g c1 c0) \to ((arity g c1 t (asucc g a)) \to ((arity g c0 u a) -\to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 -(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) +(CHead d1 (Bind Abbr) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind +Abbr) u1) H5) in (\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u +(\lambda (t0: T).(arity g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2 +(\lambda (c0: C).(arity g c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2 +(\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda +(d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 +d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a0: A).(arity g d1 u1 a0))))))))))) (\lambda (H8: (eq T u u1)).(eq_ind T u1 -(\lambda (t0: T).((csuba g c1 d1) \to ((arity g c1 t (asucc g a)) \to ((arity -g d1 t0 a) \to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) -(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A -(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) -t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda -(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: -A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a0: A).(arity g d1 u1 a0)))))))))) (\lambda (H9: (csuba g c1 d1)).(\lambda -(H10: (arity g c1 t (asucc g a))).(\lambda (H11: (arity g d1 u1 -a)).(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead -d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda -(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) -(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: -A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: -A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda -(a0: A).(arity g d1 u1 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda -(u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) +(a0: A).(arity g d1 u1 a0)))))))) H2 d1 H8) in (let H12 \def (eq_ind C c2 +(\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (or_intror (ex2 C (\lambda +(d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda +(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: +T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 -a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H9 H10 H11))))) u -(sym_eq T u u1 H8))) c2 (sym_eq C c2 d1 H7))) H6))) c H3 H4 H0 H1 H2)))]) in -(H0 (refl_equal C c) (refl_equal C (CHead d1 (Bind Abbr) u1)))))))). +a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: +A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: +C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: +C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g a0))))) (\lambda +(_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 a0)))) c1 t a +(refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10)))))))) H6)))))))))))) c y +H0))) H))))). theorem csuba_gen_flat_rev: \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall @@ -678,65 +630,57 @@ C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) \def \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda -(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(let H0 \def (match H -in csuba return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 -c1)).((eq C c0 c) \to ((eq C c1 (CHead d1 (Flat f) u1)) \to (ex2_2 C T +(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(insert_eq C (CHead +d1 (Flat f) u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda -(d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) with [(csuba_sort n) -\Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n) -(CHead d1 (Flat f) u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort -n) (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(eq C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba -g d2 d1)))))) (\lambda (H2: (eq C (CSort n) (CHead d1 (Flat f) u1))).(let H3 -\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead d1 (Flat f) u1) H2) in (False_ind (ex2_2 C T (\lambda (d2: -C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) (\lambda (d2: -C).(\lambda (_: T).(csuba g d2 d1)))) H3))) c H0 H1))) | (csuba_head c1 c2 H0 -k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda (H2: (eq C -(CHead c2 k u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 k u) (\lambda -(c0: C).((eq C (CHead c2 k u) (CHead d1 (Flat f) u1)) \to ((csuba g c1 c2) -\to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H3: -(eq C (CHead c2 k u) (CHead d1 (Flat f) u1))).(let H4 \def (f_equal C T -(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) -\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat -f) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return -(\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow -k0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C C -(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) -\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u) (CHead d1 -(Flat f) u1) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Flat f)) \to ((eq -T u u1) \to ((csuba g c1 c0) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: -T).(eq C (CHead c1 k u) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1)))))))) (\lambda (H7: (eq K k (Flat f))).(eq_ind K -(Flat f) (\lambda (k0: K).((eq T u u1) \to ((csuba g c1 d1) \to (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H8: -(eq T u u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) t) (CHead d2 (Flat -f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (H9: -(csuba g c1 d1)).(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C -(CHead c1 (Flat f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda -(_: T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Flat f) u1)) H9)) u -(sym_eq T u u1 H8))) k (sym_eq K k (Flat f) H7))) c2 (sym_eq C c2 d1 H6))) -H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow -(\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4: (eq C (CHead -c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 (Bind Abst) t) -(\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1)) \to -((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to -(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f) -u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) (\lambda (H5: -(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind -C (CHead c2 (Bind Abbr) u) (\lambda (e: C).(match e in C return (\lambda (_: +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (y: C).(\lambda (H0: +(csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1 +(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f) +u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return +(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) +\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T +(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) +(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) H2)))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2 +(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u) +(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e +in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ +_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H5 +\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) +with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k +u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e: +C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | +(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in +(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u1 +(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 +k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2: +C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda +(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (let H9 \def (eq_ind C c2 +(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda +(d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: +C).(\lambda (_: T).(csuba g d2 d1)))))) H2 d1 H8) in (let H10 \def (eq_ind C +c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (ex2_2_intro C T (\lambda +(d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) u1) (CHead d2 (Flat f) +u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u1 (refl_equal C +(CHead c1 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: +C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2 +(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq +C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 +d1))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g +a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C +(CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C +(CHead c2 (Bind Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat -_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind ((csuba -g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to (ex2_2 C T -(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 -(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H6))) -c H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c) (refl_equal C (CHead d1 (Flat -f) u1))))))))). +_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C +T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2 +(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) +H6)))))))))))) c y H0))) H)))))). theorem csuba_gen_bind_rev: \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall @@ -745,97 +689,78 @@ B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))))) \def \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda -(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(let H0 \def -(match H in csuba return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csuba -? c c0)).((eq C c c2) \to ((eq C c0 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind -b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 -e1)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) -c2)).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(eq_ind C (CSort -n) (\lambda (c: C).((eq C (CSort n) (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind -b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 -e1))))))) (\lambda (H2: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(let H3 -\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_: -C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow -False])) I (CHead e1 (Bind b1) v1) H2) in (False_ind (ex2_3 B C T (\lambda -(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind b2) -v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))) -H3))) c2 H0 H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C -(CHead c1 k u) c2)).(\lambda (H2: (eq C (CHead c0 k u) (CHead e1 (Bind b1) -v1))).(eq_ind C (CHead c1 k u) (\lambda (c: C).((eq C (CHead c0 k u) (CHead -e1 (Bind b1) v1)) \to ((csuba g c1 c0) \to (ex2_3 B C T (\lambda (b2: -B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))) -(\lambda (H3: (eq C (CHead c0 k u) (CHead e1 (Bind b1) v1))).(let H4 \def -(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with -[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u) +(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(insert_eq C +(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c2 c)) (\lambda (_: +C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 +(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: +T).(csuba g e2 e1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c2 +y)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c0 (CHead e1 (Bind +b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: +T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e2 e1)))))))) (\lambda (n: nat).(\lambda (H1: (eq +C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n) +(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) +\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) +v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda +(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda +(e2: C).(\lambda (_: T).(csuba g e2 e1))))) H2)))) (\lambda (c1: C).(\lambda +(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1 +(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda +(v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: +C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (k: K).(\lambda (u: +T).(\lambda (H3: (eq C (CHead c3 k u) (CHead e1 (Bind b1) v1))).(let H4 \def +(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with +[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | -(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead e1 (Bind b1) v1) H3) -in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda -(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) -(CHead c0 k u) (CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c: -C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((csuba g c1 c) \to (ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k u) +(CHead _ k0 _) \Rightarrow k0])) (CHead c3 k u) (CHead e1 (Bind b1) v1) H3) +in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) +(CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind +b1))).(\lambda (H8: (eq C c3 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1))))))))) (\lambda (H7: (eq K k (Bind b1))).(eq_ind K (Bind -b1) (\lambda (k0: K).((eq T u v1) \to ((csuba g c1 e1) \to (ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 u) +T).(csuba g e2 e1)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T +(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: -T).(csuba g e2 e1)))))))) (\lambda (H8: (eq T u v1)).(eq_ind T v1 (\lambda -(t: T).((csuba g c1 e1) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: -C).(\lambda (v2: T).(eq C (CHead c1 (Bind b1) t) (CHead e2 (Bind b2) v2))))) -(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) -(\lambda (H9: (csuba g c1 e1)).(let H10 \def (eq_ind T u (\lambda (t: T).(eq -C (CHead c1 k t) c2)) H1 v1 H8) in (let H11 \def (eq_ind K k (\lambda (k0: -K).(eq C (CHead c1 k0 v1) c2)) H10 (Bind b1) H7) in (let H12 \def (eq_ind_r C -c2 (\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind b1) -v1) H11) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C (CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda -(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 -(refl_equal C (CHead c1 (Bind b1) v1)) H9))))) u (sym_eq T u v1 H8))) k -(sym_eq K k (Bind b1) H7))) c0 (sym_eq C c0 e1 H6))) H5)) H4))) c2 H1 H2 -H0))) | (csuba_abst c1 c0 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C -(CHead c1 (Bind Abst) t) c2)).(\lambda (H4: (eq C (CHead c0 (Bind Abbr) u) -(CHead e1 (Bind b1) v1))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c: -C).((eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1)) \to ((csuba g c1 -c0) \to ((arity g c1 t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind -b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 -e1)))))))))) (\lambda (H5: (eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) -v1))).(let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def -(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with -[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return -(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow -Abbr])])) (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8 -\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) -with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 -(Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in (eq_ind C e1 (\lambda (c: -C).((eq B Abbr b1) \to ((eq T u v1) \to ((csuba g c1 c) \to ((arity g c1 t -(asucc g a)) \to ((arity g c u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda -(e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) -v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 -e1))))))))))) (\lambda (H9: (eq B Abbr b1)).(eq_ind B Abbr (\lambda (_: -B).((eq T u v1) \to ((csuba g c1 e1) \to ((arity g c1 t (asucc g a)) \to -((arity g e1 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda -(v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda -(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))))) (\lambda -(H10: (eq T u v1)).(eq_ind T v1 (\lambda (t0: T).((csuba g c1 e1) \to ((arity -g c1 t (asucc g a)) \to ((arity g e1 t0 a) \to (ex2_3 B C T (\lambda (b2: +T).(csuba g e2 e1)))))) (let H9 \def (eq_ind C c3 (\lambda (c: C).((eq C c +(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H8) in (let +H10 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H8) in +(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C +(CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda +(e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 (refl_equal C (CHead c1 +(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1: +C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 +(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: +C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: +B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (t: +T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u: +T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) +u) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match +e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _ +_) \Rightarrow c])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in +((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: +C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k +in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) +\Rightarrow Abbr])])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in +((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: +C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead +c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abbr +b1)).(\lambda (H10: (eq C c3 e1)).(let H11 \def (eq_ind T u (\lambda (t0: +T).(arity g c3 t0 a)) H4 v1 H8) in (let H12 \def (eq_ind C c3 (\lambda (c: +C).(arity g c v1 a)) H11 e1 H10) in (let H13 \def (eq_ind C c3 (\lambda (c: +C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: +B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) +(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 +H10) in (let H14 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H10) +in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b) +v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq +C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda +(_: T).(csuba g e2 e1))))))) H13 Abbr H9) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g -e2 e1))))))))) (\lambda (H11: (csuba g c1 e1)).(\lambda (_: (arity g c1 t -(asucc g a))).(\lambda (_: (arity g e1 v1 a)).(let H14 \def (eq_ind_r C c2 -(\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind Abst) -t) H3) in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).(csuba g (CHead c1 -(Bind Abst) t) (CHead e1 (Bind b) v1))) H14 Abbr H9) in (ex2_3_intro B C T -(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind -Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: -C).(\lambda (_: T).(csuba g e2 e1)))) Abst c1 t (refl_equal C (CHead c1 (Bind -Abst) t)) H11)))))) u (sym_eq T u v1 H10))) b1 H9)) c0 (sym_eq C c0 e1 H8))) -H7)) H6))) c2 H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c2) (refl_equal C -(CHead e1 (Bind b1) v1))))))))). +e2 e1)))) Abst c1 t (refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7)) +H6)))))))))))) c2 y H0))) H)))))).